- At , a 1D harmonic oscillator is in a linear combination of the energy eigenstates

Find the expected value of as a function of time using operator methods. - Evaluate the ``uncertainty'' in for the 1D HO ground state
where
.
Similarly, evaluate the uncertainty in for the ground state.
What is the product
?
Now do the same for the first excited state.
What is the product
for this state?
- An operator is Unitary if
.
Prove that a unitary operator preserves inner products, that is
.
Show that if the states are orthonormal, that the states are also orthonormal.
Show that if the states form a complete set, then the states also form a complete set.
- Show at if an operator is hermitian, then the operator is unitary.
- Calculate
and
.
- Calculate
by direct calculation.
Now calculate the same thing using
.
- If is a polynomial in the operator ,
show that
.
As a result of this, note that since any energy eigenstate can be written as a series
of raising operators times the ground state, we can represent by
.
- At a particle is in the one dimensional Harmonic Oscillator state
.
- Compute the expected value of as a function of time using and in the Schrodinger picture.
- Now do the same in the Heisenberg picture.

Jim Branson 2013-04-22